dice-roll-simulation

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Algorithms

A die simulator generates a random number from <code>1</code> to <code>6</code> for each roll. You introduced a constraint to the generator such that it cannot roll the number <code>i</code> more than <code>rollMax[i]</code> (1-indexed) consecutive times.

Given an array of integers <code>rollMax</code> and an integer <code>n</code>, return the number of distinct sequences that can be obtained with exact <code>n</code> rolls. Since the answer may be too large, return it modulo <code>109 + 7</code>.

Two sequences are considered different if at least one element differs from each other.

&nbsp;
Example 1:

<pre>
Input: n = 2, rollMax = [1,1,2,2,2,3]
Output: 34
Explanation: There will be 2 rolls of die, if there are no constraints on the die, there are 6 * 6 = 36 possible combinations. In this case, looking at rollMax array, the numbers 1 and 2 appear at most once consecutively, therefore sequences (1,1) and (2,2) cannot occur, so the final answer is 36-2 = 34.
</pre>

Example 2:

<pre>
Input: n = 2, rollMax = [1,1,1,1,1,1]
Output: 30
</pre>

Example 3:

<pre>
Input: n = 3, rollMax = [1,1,1,2,2,3]
Output: 181
</pre>

&nbsp;
Constraints:

<ul>
	<li><code>1 &lt;= n &lt;= 5000</code></li>
	<li><code>rollMax.length == 6</code></li>
	<li><code>1 &lt;= rollMax[i] &lt;= 15</code></li>
</ul>

dieSimulator

Dice Roll Simulation

Find Missing Observations

find-missing-observations

找出缺失的观测数据

Number of Distinct Roll Sequences

number-of-distinct-roll-sequences

不同骰子序列的数目

Array

Dynamic Programming

有一个骰子模拟器会每次投掷的时候生成一个 1 到 6 的随机数。

不过我们在使用它时有个约束，就是使得投掷骰子时，连续 掷出数字&nbsp;<code>i</code>&nbsp;的次数不能超过&nbsp;<code>rollMax[i]</code>（<code>i</code>&nbsp;从 1 开始编号）。

现在，给你一个整数数组&nbsp;<code>rollMax</code>&nbsp;和一个整数&nbsp;<code>n</code>，请你来计算掷&nbsp;<code>n</code>&nbsp;次骰子可得到的不同点数序列的数量。

假如两个序列中至少存在一个元素不同，就认为这两个序列是不同的。由于答案可能很大，所以请返回 模&nbsp;<code>10^9 + 7</code>&nbsp;之后的结果。

&nbsp;

示例 1：

<pre>输入：n = 2, rollMax = [1,1,2,2,2,3]
输出：34
解释：我们掷 2 次骰子，如果没有约束的话，共有 6 * 6 = 36 种可能的组合。但是根据 rollMax 数组，数字 1 和 2 最多连续出现一次，所以不会出现序列 (1,1) 和 (2,2)。因此，最终答案是 36-2 = 34。
</pre>

示例 2：

<pre>输入：n = 2, rollMax = [1,1,1,1,1,1]
输出：30
</pre>

示例 3：

<pre>输入：n = 3, rollMax = [1,1,1,2,2,3]
输出：181
</pre>

&nbsp;

提示：

<ul>
	<li><code>1 &lt;= n &lt;= 5000</code></li>
	<li><code>rollMax.length == 6</code></li>
	<li><code>1 &lt;= rollMax[i] &lt;= 15</code></li>
</ul>